Reeb spaces of continuous real-valued functions on topological spaces are fundamental and strong tools in investigating the spaces. The Reeb space is the natural quotient space of the space of the domain represented by connected components of its level sets. They have appeared in theory of Morse functions in the last century and as important topological objects, they are shown to be graphs for tame functions on (compact) manifolds such as Morse(-Bott) functions and naturally generalized ones. Related general theory develops actively, recently, mainly by Gelbukh and Saeki. For nice Haudorff spaces and continuous functions there, they are "$1$-dimensional". We concentrate on Reeb spaces which are not CW complexes and study their representations by graphs and nice examples. Reconstructing nice smooth functions with given Reeb graphs is of related studies and pioneered by Sharko and followed by Masumoto, Michalak, Saeki, and so on. The author has also contributed to it.

We determine all orthogonal transformations of a quadratic space over reals such that any orthogonal transformation which is conjugate to one of them in the linear group is conjugate in the orthogonal group.

Ball lightning remains one of the most enigmatic atmospheric phenomena, characterized by its long lifetime, ability to penetrate materials, and stable spherical structure. Here we propose a novel theoretical framework interpreting ball lightning as a three-dimensional projection of a high-dimensional topological soliton. The system is described by a nonlinear Schrödinger equation with attractive interactions, stabilized by a non-zero topological charge. Through comprehensive numerical simulations of the three-dimensional Gross-Pitaevskii equation, we verify the model's core predictions: (1) long-lived stability protected by topological invariants, (2) low transmission probability due to wavefunction orthogonality, and (3) energy and size scales consistent with observational data. The soliton lifetime $τ\sim \hbar/Γ$ naturally explains the observed second-scale durations. Our work provides a self-consistent physical explanation for ball lightning while offering concrete pathways for experimental realization of three-dimensional topological solitons in Bose-Einstein condensates and nonlinear optical systems. This theoretical framework gains additional support from recent experimental breakthroughs in laboratory generation of ball-lightning-like structures.

We present the computation of the next-to-leading order QCD corrections to the $pp\to t\bar{t} H+X$ process in the di-lepton channel at the LHC, including relevant dimension-6 operators $({\cal O}_{tϕ}, \, {\cal O}_{ϕG},\, {\cal O}_{tG}, \, {\cal O}_{tW})$ from the Standard Model Effective Field Theory. In our studies, higher-order corrections and effective operators are consistently included in the production part of the process as well as in the top-quark decays. We perform a detailed study of linear, cross, and quadratic contributions and their uncertainties, including renormalisation group effects. Our findings are presented at the integrated and differential cross-section level for the LHC Run III center-of-mass energy of $\sqrt{s}=13.6$ TeV. Finally, we provide predictions for $pp\to t\bar{t} H+X$ with stable top quarks and compare them with the results in which top quarks are reconstructed from their decay products. We show that kinematic cuts, as well as higher-order effects and SMEFT operators in top-quark decays, are important and should be consistently considered together, because they have a significant impact on the shape of the standard observables measured for the $pp\to t\bar{t}H+X$ process at the LHC.

Let $f(d)$ be the smallest value for which every bridgeless graph $G$ with diameter $d$ admits a strong orientation $\overrightarrow{G}$ such that the diameter of $\overrightarrow{G}$ is at most $f(d)$. Chvátal and Thomassen (JCTB, 1978) established general bounds for $f(d)$, and also proved that $f(2)=6$ and $f(4)\geq 12$. The works of both Kwok, Liu and West (JCTB, 2010) and Wang and Chen (JCTB, 2022) together determined $f(3)=9$. In this paper, we improve the best known upper bound for $f(4)$ from $21$ (Babu et al., DAM, 2021) to \textbf{$18$}.

We consider the problem of reaching consensus in communication networks that are modeled by directed graphs. We assume the existence of a message authentication mechanism (such as digital signatures) to verify the integrity of messages. We identify the necessary and sufficient conditions on the directed communication graph for the following problems to be solvable: (i) exact consensus in synchronous systems; and (ii) approximate consensus in asynchronous systems.

The goal of this article is twofold. We introduce a notion of convergence for Lorentzian pre-length spaces, $\ell$-convergence, that extends previous convergence notions in this context. We show that timelike curvature and timelike curvature-dimension bounds are stable under (measured) $\ell$-convergence. Then, we show that $\ell$-convergence is well adapted for generalized Lorentzian cones: a sequence of generalized cones $-I_i\times_{f_i}X_i$ converges in $\ell$ sense if the base $I_i$ and the fiber $X_i$ converge in GH sense and the functions $f_i$ converge uniformly. We use this to show sharp timelike curvature and timelike curvature-dimension bounds for such cones. Finally, we obtain a pre-compactness theorem for $\ell$-convergence in the class of smooth generalized cones that have a uniform lower bound on the full Ricci (or Riemann) curvature tensor.

Optical skyrmionic textures, including Néel-type skyrmions and merons, have attracted significant attention across diverse physical systems for their promising applications in ultra-precise metrology, optical information processing, and quantum technologies. However, the lack of an effective approach for their on-chip directional transport and manipulation impedes their applications in photonic integrated devices. Here, we numerically demonstrate a photonic platform that utilizes topologically protected valley edge state to achieve robust on-chip directional transport of optical spin merons. These merons originate from spin-orbit coupling within the evanescent field at the valley photonic crystal surface and exist as eigenstates of the topologically protected edge state, ensuring their robust propagation even in the interface with defects. Leveraging the valley degree of freedom of topological edge states, we further achieve valley-locked spin merons, enabling flexible control over the polarity of spin merons. By endowing spin merons with topological protection in momentum space, our work provides an approach for robust on-chip transport and manipulation of spin merons, thereby paving the way for expanding their application in photonic systems.

In this paper, we study 4-dimensional complete noncompact manifolds (M,g) satisfying Rm(g) $\in\mathfrak{C}_{η,μ}$ via Ricci flow. Under the additional assumption of maximal volume growth, we prove topological and geometric gap theorems. We also study 4-dimensional complete manifolds satisfying a lower bound with respect to $\mathfrak{C}_{η,μ}$ and obtain regularity results for Gromov-Hausdorff limits of complete volume non-collapsed manifolds satisfying such curvature lower bounds.

High-fidelity (HF) data are often expensive to collect and therefore scarce, making conditional quantiles difficult to estimate accurately. We propose a two-stage, model-agnostic method for multi-fidelity quantile regression. The central idea is a local quantile link: at each covariate value, the HF quantile is represented as a low-fidelity (LF) quantile evaluated at a covariate-dependent level. This reformulation reduces the problem to estimating the level function, which can be smoother than the HF quantile itself when the LF and HF conditional distributions have similar shapes. We also study the complementary regime in which this advantage weakens and introduce a correction step to improve robustness. Our theory characterizes when the proposed estimator converges faster than direct quantile regression using HF data alone and when the correction step provides further improvement. Experiments on synthetic and real data show that our method yields more accurate quantile estimates and tighter conformal prediction intervals.

The Jarzynski equality is a cornerstone of nonequilibrium thermodynamics, linking work statistics to equilibrium free-energy differences. Although it has been extensively verified in classical and quantum Hermitian settings, its status in non-Hermitian dynamics remains under debate. Here we show that, in a postselected no-quantum-jump framework, a conditional non-Hermitian Jarzynski equality holds when the transition probabilities obey a parity-exchange symmetry. We study a constructed family of two-level hybrid Hamiltonians formed as linear combinations of parity-time ($\mathcal{PT}$) and anti-parity-time ($\mathcal{APT}$) symmetric terms, and demonstrate using complementary geometric and algebraic arguments that the parity-exchange symmetry persists throughout the corresponding $\mathrm{SU}(2)$-rotated orbit. Relative to previous $\mathcal{PT}$-focused conditional Jarzynski equality results, the advance here is an extension of the symmetry criterion from the isolated $\mathcal{PT}$ endpoint to a broader $\mathcal{PT}$--$\mathcal{APT}$ hybrid family. Experimentally, we implement three representative points, $θ_k = 0, π/4, π/2$, in a single trapped $^{171}\mathrm{Yb}^+$ ion and measure the resulting work distributions under cyclic protocols with $ΔF = 0$, confirming the predicted symmetry criterion at those points. Our results establish a symmetry-based extension of the conditional non-Hermitian Jarzynski relation within this restricted two-level setting.

This paper introduces a Lie algebra analogue of the middle convolution functor, which is defined on the category of modules over certain Lie algebras, including, as typical motivating examples, free Lie algebras, Drinfeld-Kohno Lie algebras, and the holonomy Lie algebras of complements of hyperplane arrangements. First, we demonstrate that the middle convolution for Lie algebra representations can be regarded as a natural generalization of the infinitesimal analogue of the Long-Moody functor for Drinfeld-Kohno Lie algebras. Second, we show that our middle convolution recovers the Dettweiler-Reiter additive middle convolution for Fuchsian systems on the punctured Riemann sphere as a special case. Furthermore, we show that when applied to the holonomy Lie algebra of the complement of a hyperplane arrangement, our functor is compatible with Haraoka's middle convolution for logarithmic connections on such complements. Finally, we establish a Riemann-Hilbert correspondence between the middle convolution for the holonomy Lie algebra and the middle convolution for local systems on complements of hyperplane arrangements.

X-ray absorption spectroscopy (XAS) and resonant inelastic X-ray scattering (RIXS) are powerful probes of electronic structure owing to their chemical and orbital selectivity. For powder samples, however, interpreting RIXS spectral intensities remains challenging as the measured signal is an average over all orientations. Existing theoretical treatments rely largely on spherical-tensor formalisms, which often involve complex derivations and case-specific analyses. Meanwhile, recent advances in quantum-chemistry methods have made the evaluation of transition tensors in Cartesian coordinates both accurate and straightforward. Here, we present a general theoretical framework that translates Cartesian transition tensors into physically meaningful, orientation-averaged intensities for powder samples. The formalism allows predicting angular and polarization dependences \textit{ab initio} for both XAS and RIXS and is extendable to other spectroscopies. The resulting predictions show excellent agreement with RIXS experimental data at the Ce L$_3$ edge.

Many economic interventions are designed as marginal changes in incentives. Yet in environments shaped by coordination, institutional persistence, and path dependence, such reforms often leave behavior largely unchanged. This paper studies interventions in games when equilibrium selection displays status-quo inertia: if the pre-intervention equilibrium remains a Nash equilibrium after policy, it continues to be selected. In that environment, price-based interventions and simple option expansion may fail even when they improve welfare in a partial-equilibrium sense. By contrast, interventions that modify the feasible action space, especially deletion and replacement interventions, can be substantially more effective because they remove the strategic basis for persistence. We develop a simple framework, derive general results, provide complete proofs, and illustrate the economics with examples from climate transition, platform regulation, financial reform, and industrial modernization. The analysis highlights a basic policy lesson: when inefficient equilibria are institutionally entrenched, the central problem is often not how to price the existing game more finely, but how to change the game itself.

We study distributional Bernoulli-type conditions in geometrically irregular domains $Ω$. Here the zero extension of $u$ to $\mathbb{R}^n$ satisfies $$ Δu \;=\; c\,\mathcal{H}^{n-1}\!\lfloor_{\partial^*Ω}\;-\;f(u)\,\mathbf{1}_Ω\,dx $$ in the distributional sense. This is a weak version of the one-phase Bernoulli free boundary condition, which avoids the uniform Lipschitz/density assumptions of classical Alt--Caffarelli theory. We prove that for every $n \ge 2$ and every $f \in C^2(\mathbb{R})$ with $f(0)>0$, there exist bounded, non-spherical, finite-perimeter domains $Ω\subset \mathbb R^n$ satisfying this distributional Bernoulli law with $$ 0<\mathcal H^{n-1}(\partial^*Ω)<\infty, \qquad \mathcal H^{n-1}(\partialΩ\setminus\partial^*Ω)=0, $$ yet $$ {\rm ess}\sup_{x\in\partial^*Ω} \sup_{0<r<1} \frac{\mathcal H^{n-1}(B_r(x)\cap\partial^*Ω)}{r^{n-1}} =\infty. $$ This shows the key constraint is not absence of a reduced boundary, but failure of uniform all-scale surface density bounds. For $f \equiv 1$, these results yield counterexamples to the weak Serrin-type overdetermined problems in all dimensions, proving the distributional Bernoulli law alone cannot replace the uniform growth/density conditions core to Alt--Caffarelli theory. On the other hand, we prove a planar rigidity result: Within the Smirnov class, the associated harmonic quadrature identity forces $Ω$ to be a disk. Thus, for the constant-source Serrin/Bernoulli law, Smirnov regularity is the threshold for weak Bernoulli rigidity in $\mathbb R^2$, while uniform upper density bounds form the threshold for $n\ge 3$ according to [23].

In this work, a unified Lie-algebraic formulation of catability is constructed for relativistic quantum systems with arbitrary spin within this framework. In this case, the analysis starts with constructing catability as a quantitative measure for superposed coherent states, where coherence structure and quantum interference properties are studied using algebraic representations in this framework. Also, a generalized Foldy-Wouthuysen transformation is formulated within a Lie algebraic framework, delivering a systematic procedure for block-diagonalization of relativistic Hamiltonians and separation of positive- and negative-energy components in this framework. Within this formalism, a phase-sensitive catability operator is introduced to study phase correlations and coherence effects in the relativistic quantum dynamics framework. The approach is applied to Dirac spin-$1/2$ particles, where relativistic fermionic catability is analyzed in relation to spinorial structures and symmetry generators framework. The formalism is extended through a unified geometric and Lie-algebraic treatment, establishing a consistent description of catability in a relativistic quantum mechanics framework. In this context, the generalized framework is constructed for arbitrary spin-$s$ fields, enabling investigation of higher-spin relativistic quantum states within the same algebraic structure framework. In this context, the obtained results show a generalized theoretical platform for investigating relativistic quantum coherence, superposition effects, and algebraic symmetries in the framework of fermionic and bosonic systems.

We determine the complexity of exact evaluation of the Reshetikhin--Turaev and Turaev--Viro invariants of closed connected oriented 3-manifolds, with the underlying tensor category fixed. If $\mathcal C$ is a modular category, then the Reshetikhin--Turaev invariant $Z_{\mathcal C}(M)$ can be computed in polynomial time from a framed-link surgery presentation of $M$ precisely when $\mathcal C$ is pointed; otherwise the problem is $\#\mathrm{P}$-hard. If $\mathcal A$ is a spherical fusion category, then the Turaev--Viro invariant $|M|_{\mathcal A}$ can be computed in polynomial time from a triangulation of $M$ precisely when the Drinfeld center $\mathcal Z(\mathcal A)$ is pointed, equivalently when $\mathcal A$ is trivializable pointed; otherwise the problem is $\#\mathrm{P}$-hard. This proves the dichotomy conjectured by Bridges and Samperton and identifies the categorical obstruction to polynomial-time evaluation.

We give a microlocal description of the Aubert--Zelevinsky involution for all unipotent representations of all inner forms of simple adjoint unramified $p$-adic groups. Via the realization of enhanced $L$-parameters as perverse sheaves, we show that the involution corresponds to the composition of three operations on an endoscopic subgroup: Fourier transform, Chevalley involution and duality on local systems. When the group is not inner to an unramified triality form of $D_4$ we further show that one does not need to pass to an endoscopic subgroup. This was previously verified in certain special examples by several authors where only the contribution by Chevalley involution and Fourier transform was observed. Duality on local systems is invisible in those examples since only self-dual local systems appear. Motivated by categorical considerations, we provide a second formulation, involving complex conjugation from the compact form of the dual group, giving a covariant functor of perverse sheaves that agrees with Aubert--Zelevinsky duality on $L$-parameters and as involutions of graded Hecke algebras. This formulation holds without passing to an endoscopic subgroup and is valid for all inner forms of simple adjoint unramified groups. Finally, we prove the microlocal Hiraga conjecture for unipotent $A$-parameters of inner-to-split simple adjoint groups as a consequence of our results.

Support vector regression with Mean Absolute Percentage Error (MAPE) loss is theoretically well-motivated for forecasting applications where accuracy is evaluated in relative terms, but the sample-dependent dual box constraints it induces have not been addressed in the published SMO literature. We derive a Sequential Minimal Optimization algorithm for this setting and prove a structural-invariance result: the MAPE modification affects exactly two components of the SMO iteration -- working-set selection and analytic-update clipping -- leaving gradient bookkeeping and curvature computation identical to classical epsilon-SVR. Building on this invariance, we establish four efficiency improvements (asymmetric freeze-counters, warm-starting, block working-set updates of size four, and per-pair tolerance scaling) and resolve a previously-open convergence problem for the odd-symmetry kernel variant via adaptive spectral regularization. Numerical validation against three reference solvers across eleven synthetic configurations certifies solution agreement within standard tolerance. Wall-time benchmarks show the present algorithm achieves the lowest median runtime on every tested configuration against OSQP, MOSEK, and Clarabel. At production scale, the algorithm converges on the California Housing benchmark while the patched LIBSVM reference implementation reaches its iteration ceiling without satisfying optimality -- demonstrating the practical necessity of the theoretical efficiency mechanisms. An open-source R package and an explicit solver-adaptation recipe are provided.

We characterize decomposable principally polarized abelian varieties of the form $E\times B$, with $E$ an elliptic curve, in two different ways, which are, surprisingly, completely analogous to classical results of curve theory concerning hyperelliptic curves. The first one is by the failure of a normal generation property, namely the generation in degree zero of a certain graded module over the symmetric algebra over $H^0(2Θ)$. This appears to be the first result of this type in the realm of p.p.a.v.'s. The second characterization is by the failure of surjectivity of second order gaussian maps associated to line bundles corresponding to $6Θ$, or, equivalently, by the fact that at some point, the line bundle corresponding to $3Θ$ fails to separate $2$-jets. We also show that this last result is equivalent to an effective version of a theorem of Nakamaye characterizing the above decomposable abelian varieties as those computing the minimal Seshadri constant. Finally we propose some conjectural generalizations relating $p$-jets separation thresholds, higher gaussian maps sujectivity thresholds, and Seshadri constants.

The primary objective of this work is the determination of fluences and characteristics of fast neutrons, other hadrons, and highly ionizing particles in the environment of the MoEDAL experiment at the Large Hadron Collider. These particles constitute an experimental background for the passive Nuclear Track Detectors (NTDs) used by MoEDAL to search for tracks potentially produced by Dirac magnetic monopoles, in particular by particles indistinguishable in NTD from monopoles. The study is based on data acquired by the Timepix hybrid silicon pixel detector network, which represents the first and only active detector system installed and operated as part of the MoEDAL experiment from 2013 to 2018. The Timepix detector network enables real-time measurements of mixed radiation fields, including the composition, spectral properties, and directional characteristics of individual radiation components across different regions of the MoEDAL experimental area. The paper presents detailed results of the radiation field analysis with emphasis on neutrons and highly ionizing particles, including their directional distributions. The first results demonstrating the spatial tracking capabilities of the Timepix detectors are also reported, illustrating the reconstruction of particle direction and energy-loss profiles from individual detector frames.

Local projections (LP) and vector autoregressions (VAR) are the two standard tools for impulse response analysis, but they often display a finite-sample trade-off: LP is typically less biased but more volatile, while VAR is more precise but can be biased under misspecification. We propose an easy-to-implement estimator-averaging approach that combines LP and VAR at each horizon by minimizing the mean squared error of the impulse response itself, rather than in-sample fit. We derive closed-form oracle weights for this finite-sample risk problem, develop feasible AR-sieve-bootstrap procedures, and compare them against an Rsquare-based model-averaging benchmark. For a benchmark class of short-memory linear data generating processes in which LP and VAR are both consistent, we establish the consistency and limiting distribution of the feasible averaged estimator. Monte Carlo results show meaningful risk reductions relative to LP and VAR alone. In an empirical application revisiting Bauer and Swanson (2023), estimator averaging delivers stable and economically intuitive responses for yields, activity, prices, and credit spreads.

Well-known chaotic maps, such as the logistic and tent maps, have been used to generate cryptographically secure pseudorandomness, yet we know of no efforts which attempt to use the Gauss continued-fraction map, a known chaotic map, as a starting point for producing quality pseudorandom output. In this paper, we consider the family of $r$-continued-fraction maps, which generalize the Gauss map, and use them to generate pseudorandom output which outperforms many standard generators, such as the Mersenne Twister, in statistical quality, as ascertained by use of the Dieharder, PractRand, and TestU01 suites. In this way, we demonstrate the potential viability of these maps as a starting point for novel generators, and provide practical motivation for further study of the properties of both the exact and finite-precision $r$-continued fraction maps.

A renormalization-group (RG) improved Schwarzschild-like black hole (BH) is studied here, with a lapse that interpolates between a classical Schwarzschild exterior and a quantum-smoothed interior set by a cutoff scale $ξ$ and an interpolation parameter $γ$. We work out the horizon structure together with the photon sphere and shadow radius $R_{\mathrm{sh}}$, set up the scalar, electromagnetic, and Dirac Regge-Wheeler-Zerilli problems in a single treatment, and compute the fundamental and overtone quasinormal modes by sixth-order WKB, cross-checked against time-domain ringdown. For $ξ>0$ and $γ>0$ the geometry is regular, with a de Sitter core. Strong Cosmic Censorship (SCC) is examined at the inner Cauchy horizon, which the improved geometry generates without charge or rotation. The quasinormal spectral gap $β=|\mathrm{Im}\,ω|/κ_-$ stays multipole-independent at the $6\%$ level and follows $β\simeqλ_{L}/(2κ_{-})$. It remains below the de Sitter Christodoulou bound across the parameter range, and the asymptotically flat late-time tail places the geometry in the SCC-respecting class. A thermodynamic analysis identifies a Davies-type phase transition of the outer horizon, with the Schwarzschild $T_{H}\propto 1/r_{+}$ decay replaced by a bell-curve profile peaking at $T_{H}^{\max}\simeq 0.062$. A scan of the $(ξ,γ)$ plane gathers the joint behavior of the shadow, the scalar barrier, the SCC ratio, and $T_H$. Set against Bardeen, Hayward, and Bonanno-Reuter BHs at matched perturbation scale, the improved Schwarzschild BH is the most Schwarzschild-like of the regular-BH family, its static shadow radius degenerate with Hayward and Bonanno-Reuter at the percent level. The closing analysis takes up the sparsity of the Hawking flux and the energy-emission rate, both tied to the outer-horizon surface gravity through a single auxiliary function.

The simultaneous observation of strong sequential suppression and small azimuthal anisotropy for bottomonium states provides a stringent constraint on the time scale and geometry of the suppression mechanism. We propose an early-time geometric survival mechanism in which the strong pre-equilibrium color field induces a local proper acceleration and an associated Unruh causal scale. The survival probability is modeled by a WKB-motivated exponential controlled by the ratio of the quarkonium radius to the causal horizon. Combined with CGC-inspired centrality and energy scaling, the framework gives a compact description of the $Υ(1S, 2S, 3S)$ suppression pattern and predicts a definite RHIC/LHC energy dependence. Because the suppression acts as a local scalar factor before hydrodynamic response develops, it naturally produces little additional quarkonium momentum anisotropy.

We propose a simple estimator for the dynamic decomposition of the Generalized Dynamic Factor Model that avoids frequency-domain methods. First, we show that it is a reasonable approximation to assume that the dynamic common component of the Generalized Dynamic Factor Model admits a representation in terms of current and lagged statically pervasive factors. Then, assuming finite lag order, this simplification reduces estimation to a regression of the observed variables on estimated factors and their lags, where the factors are extracted via static principal components. The proposed approach naturally accommodates weak, non-pervasive factors within the dynamic common space. We establish consistency and asymptotic normality for both the dynamic and weak common components under a new asymptotic framework that allows for such weak factors. In an application to three high-dimensional time series panels of European macroeconomic data we detect a sizeable weak common component share in several key macroeconomic indicators.

We study stochastic nonconvex-concave minimax optimization with nonlinear coupled constraints that are convex in the maximization variable. To address the nonsmoothness arising from such constraints, we develop a penalty-based smooth approximation that combines quadratic penalization of the coupled constraints with quadratic regularization of the inner maximization problem. Based on this approximation, we propose SPACO, a single-loop stochastic gradient algorithm that tracks the inner maximizer by one stochastic ascent step, updates the outer variable using an inexact stochastic descent direction, and adaptively updates the penalty and regularization parameters over the iterations. For the penalty-based smooth approximation, we establish convergence guarantees from both minimizer and stationarity perspectives. In particular, we introduce enhanced KKT conditions and show that stationary points of the smooth approximations can converge to points satisfying these conditions. An example illustrates that the enhanced KKT conditions can help exclude KKT points that are not local minimizers. For SPACO, we prove non-asymptotic complexity bounds for stationarity and feasibility, as well as asymptotic subsequential convergence to enhanced KKT points. Numerical experiments on synthetic examples, fairness-aware classification, and constrained generative adversarial network training demonstrate the effectiveness of the proposed method.

In practical physical systems, modeling assumptions of Gaussianity and basis independence break down due to self-interactions. We study a specific instance of one-dimensional $ϕ^4$ theory on a lattice, analyzing how the interaction strength and system size jointly affect the marginal and joint distributions of frequency-based representation of the field (i.e., Fourier modes). We find that models relying on Gaussian and independent Fourier modes fail primarily from structured dependencies rather than marginal non-Gaussianity, since individual modes become approximately Gaussian despite mode coupling growing with size. Based on this, we identify three distinct regimes that delineate where traditional methods remain effective and where more expressive models are needed. Our results provide a computationally simple diagnostic to establish when Gaussian models are insufficient, and establish a concrete design criterion that future nonlinear models must satisfy.

This paper proposes specular differentiation in one-dimensional Euclidean space and provides its fundamental analysis, including a quasi-Fermat theorem and a quasi-Mean Value Theorem. As an application, this paper develops several numerical schemes for solving initial value problems for first-order ordinary differential equations. Based on numerical simulations, we select one scheme and prove its second-order consistency and convergence. By modifying this scheme, we also obtain a numerical scheme with zero local truncation error for ODEs whose solution trajectories are ellipses.

The coherent equalization problem consists in designing a quantum system acting as a mean-square near-optimal filter for a given quantum communication channel. The paper develops an improved method for the synthesis of transfer functions for such equalizing filters, based on a linear quantum system model of the channel. The method draws on a connection with the two-disk problem of ${H}_{\infty}$ control for classical (i.e., non-quantum) linear uncertain systems. Compared with the previous methods, the proposed method applies to a broader class of linear quantum communication channels.